On the equations of Poizat and Li\'enard

TL;DR

This paper analyzes the solution structures of Poizat and Lie9nard differential equations, providing criteria for strong minimality, classifying algebraic relations, and exploring solutions in special cases.

Contribution

It offers a complete classification of algebraic relations for strongly minimal Poizat equations and criteria for strong minimality, advancing understanding of these differential equations.

Findings

Necessary and sufficient condition for strong minimality.

Complete classification of algebraic relations for solutions.

Analysis of non-strongly minimal cases and applications.

Abstract

We study the structure of the solution sets in universal differential fields of certain differential equations of order two, the Poizat equations, which are particular cases of Li\'enard equations. We give a necessary and sufficient condition for strong minimality for equations in this class and a complete classification of the algebraic relations for solutions of strongly minimal Poizat equations. We also give an analysis of the non strongly minimal cases as well as applications concerning the Liouvillian and Pfaffian solutions of some Li\'enard equations.